Kane has been ruled out since New Year's Day after suffering a ruptured tendon away at Southampton.
Despite the club's initial prognosis that Kane would return to training in April, Mourinho has cast doubt on these claims.
And the former Chelsea and Manchester United boss says the England captain might not even be fit for Euro 2020.
The Three Lions' opening match of the tournament is against Croatia at Wembley on June 14.
Mourinho says he hopes Kane will be available for their Premier League run-in from the start of May but says he will not be rushed to return to the pitch.
"The match against Leicester, maybe we'll need that match for something - to be fourth, to be fifth, to be sixth, maybe we need that match for something and maybe he can help us in that match, I hope," he said.
"There is no setback, nothing is happening, everything is okay. He's having his treatment, no pressure, he keeps going, there's no setback.
"I just think that my feeling with [Hugo] Lloris is we were all speaking about February and he came back for January. It's better not to put any kind of expectations, any kind of pressure, let it go and let's see."
Kane last week published a video of himself doing hamstring strengthening exercises and has travelled to the Caribbean for warm weather training to aid his recovery, fuelling optimism about his progress.
But a return sooner than April now seems to be way off the mark.
Asked if there would be concerns for Kane to play for England at Euro 2020 after only playing one or two games for Spurs on his comeback, Mourinho suggested he won't be thinking about the striker's summer.
"That's a problem for [Gareth] Southgate and for Steve Holland, not for me," he said.
Spurs welcome Southampton to the Tottenham Hotspur Stadium for their FA Cup fourth round replay tonight.